12 problems

Let $V(m,n)$ and $V_d(m,n)$ denote the rank functions associated with the compositions considered in the paper, with $m\in\mathbb Z$ and $n\geq 0$. Zhou's unimodality conjecture. F…

For a partition of $n$, let its Dyson rank be its largest part minus its number of parts, and let $N(a,c,n)$ denote the number of partitions of $n$ whose rank is congruent to $a$ m…

Let $N(r,2;n)$ denote the number of partitions of $n$ whose rank is congruent to $r$ modulo $2$, for $r\in\{0,1\}$. Hou–Jagadeesan's conjecture. If $r=0$ (respectively, $r=1$), the…

Hou–Jagadeesan's maximal-value conjecture. The following are true:

Let $N(m,n)$ and $M(m,n)$ denote, respectively, the rank and crank counting functions for ordinary partitions of $n$ with statistic $m$. Strict log-concavity conjecture. The follow…

Wei–Zhang inequalities. For $n\geq 11$,

Ji–Zhang–Zhao inequalities. For $n\geq 0$ and $1\leq i\leq 4$,

Let $\overline{N}(m,n)$ denote the number of overpartitions of $n$ with $D$-rank $m$, and let $\overline{N2}(m,n)$ denote the number of overpartitions of $n$ with $M_2$-rank $m$. C…

Let $N(r,t;n)$ denote the number of partitions of $n$ whose Dyson rank is congruent to $r$ modulo $t$. General convexity conjecture. If $0\leq r<t$ and $t\geq2$, then … for suffici…

Let $N(r,t;n)$ be the number of partitions of $n$ with Dyson rank congruent to $r$ modulo $t$, and let $\operatorname{maxN}(r,t;n)$ be the maximum number of partitions in a rank re…

Let $N(r,t;n)$ denote the number of partitions of $n$ whose Dyson rank is congruent to $r$ modulo $t$. Convexity conjecture. If $t=2$ and $r=0$ (respectively, $r=1$), then … for al…

Let $N(m,c,n)$ denote the number of partitions of $n$ whose rank is congruent to $m$ modulo $c$. Lewis's congruence. For every integer $n\geq 0$, one has … This conjecture of Richa…